cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A379500 Square array A(n, k) = A249670(A246278(n, k)), read by falling antidiagonals; A249670(n) = A017665(n)*A017666(n), applied to the prime shift array.

Original entry on oeis.org

6, 28, 12, 2, 117, 30, 120, 40, 775, 56, 45, 1080, 1680, 2793, 132, 21, 672, 19500, 7392, 16093, 182, 84, 390, 3960, 137200, 24024, 30927, 306, 496, 176, 43400, 208, 1948584, 55692, 88723, 380, 78, 9801, 5460, 368676, 40392, 5228860, 116280, 137541, 552, 210, 9300, 488125, 17136, 2928926, 69160, 25645860, 209760, 292537, 870
Offset: 1

Views

Author

Antti Karttunen, Jan 02 2025

Keywords

Examples

			The top left corner of the array:
k=|   1      2      3        4      5        6      7          8        9       10
2k|   2      4      6        8     10       12     14         16       18       20
--+---------------------------------------------------------------------------------
1 |   6,    28,     2,     120,    45,      21,    84,       496,      78,     210,
2 |  12,   117,    40,    1080,   672,     390,   176,      9801,    9300,    6552,
3 |  30,   775,  1680,   19500,  3960,   43400,  5460,    488125,   83790,  102300,
4 |  56,  2793,  7392,  137200,   208,  368676, 17136,   6725201,   18392,   10374,
5 | 132, 16093, 24024, 1948584, 40392, 2928926, 50160, 235793305, 4082364, 4924458,

Links

Crossrefs

Elementwise product of arrays A341605 and A341606.
Cf. A246278, A249670, A355925, A379499.
Cf. A036690 (leftmost column), A361468 (even bisection gives row 2).

Programs

  • PARI
    up_to = 55;
A249670(n) = { my(ab = sigma(n)/n); numerator(ab)*denominator(ab); };
A246278sq(row,col) = if(1==row,2*col, my(f = factor(2*col)); for(i=1, #f~, f[i,1] = prime(primepi(f[i,1])+(row-1))); factorback(f));
A379500sq(row,col) = A249670(A246278sq(row,col));
A379500list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A379500sq(col,(a-(col-1))))); (v); };
v379500 = A379500list(up_to);
A379500(n) = v379500[n];

Formula

A(n, k) = A341605(n, k) * A341606(n, k).
A(n, k) = A379499(n, k) / (A355925(n, k)^2).

A178398 a(n) = p(p+1)^2, where p is the n-th prime.

Original entry on oeis.org

18, 48, 180, 448, 1584, 2548, 5508, 7600, 13248, 26100, 31744, 53428, 72324, 83248, 108288, 154548, 212400, 234484, 309808, 368064, 399748, 505600, 585648, 720900, 931588, 1050804, 1114048, 1248048, 1318900, 1468548, 2080768, 2282544, 2609028, 2724400, 3352500, 3488704, 3919348, 4384048, 4713408, 5237748, 5799600, 5995444, 7041024, 7263748, 7723188, 7960000, 9483184, 11189248, 11800368, 12114100
Offset: 1

Views

Author

Eduard Mayer, Dec 21 2010

Keywords

Links

Programs

  • Mathematica
    Table[p (p+1)^2,{p,Prime[Range[50]]}] (* Harvey P. Dale, Aug 12 2025 *)
  • PARI
    a(n) = prime(n)*(prime(n)+1)^2 \\ Michel Marcus, Jun 24 2013

Formula

a(n) = A000040(n)*A110833(n) = A000040(n)*A008864(n)^2 = A008864(n)*A036690(n). - Michel Marcus, Jun 24 2013
Sum 1/a(n) = A382567. - R. J. Mathar, Mar 31 2025

Extensions

Edited by N. J. A. Sloane, Dec 21 2010

A330397 Composite numbers k such that gpf(k)^2 + gpf(k) == 0 (mod k), where gpf(k) = A006530(k) is the greatest prime dividing k.

Original entry on oeis.org

6, 10, 12, 14, 15, 22, 26, 28, 30, 33, 34, 38, 44, 46, 51, 56, 58, 62, 66, 69, 74, 76, 82, 86, 87, 91, 92, 94, 95, 102, 106, 118, 122, 123, 124, 132, 134, 138, 141, 142, 145, 146, 153, 158, 159, 166, 172, 174, 177, 178, 182, 184, 188, 190, 194, 202, 206, 213, 214, 218, 226, 236
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Feb 25 2020

Keywords

Comments

All terms are in either A036690 or A064052. - Charles R Greathouse IV, Mar 27 2020

Links

Crossrefs

Supersequence of A000396 and A036690.
Cf. A000043, A000668, A002808, A006530, A329534, A064052.

Programs

  • Magma
    [k: k in [4..240] | -Maximum(PrimeDivisors(k))^2 mod k eq Maximum(PrimeDivisors(k))];
  • Mathematica
    Select[Range[250], (g = FactorInteger[#][[-1, 1]]) < # && Divisible[g^2 + g, #] &] (* Amiram Eldar, Feb 25 2020 *)
  • PARI
    gpf(n,f=factor(n))=f=f[,1]; if(#f==0,1,f[#f]);
is(n)=my(g=gpf(n)); gCharles R Greathouse IV, Mar 26 2020
  • PARI
    list(lim)=my(v=List()); forfactored(N=6,lim\1, my(n=N[1],f=N[2][,1],i=#f); if(i>1 && (f[i]^2+f[i])%n==0, listput(v,n))); Vec(v); \\ Charles R Greathouse IV, Mar 27 2020

A079142 Numbers divisible by prime integer parts of their square roots.

Original entry on oeis.org

4, 6, 8, 9, 12, 15, 25, 30, 35, 49, 56, 63, 121, 132, 143, 169, 182, 195, 289, 306, 323, 361, 380, 399, 529, 552, 575, 841, 870, 899, 961, 992, 1023, 1369, 1406, 1443, 1681, 1722, 1763, 1849, 1892, 1935, 2209, 2256, 2303, 2809, 2862, 2915, 3481, 3540, 3599
Offset: 0

Views

Author

Cino Hilliard, Dec 26 2002

Keywords

Comments

n is in the sequence if r=floor(sqrt(n)) is prime and r divides n.
Union of the 3 sequences A001248={p^2}, A036690={p(p+1)} and {p(p+2)} for p prime.
The sum of the reciprocals = 1.04...

Examples

			56 is in the sequence because floor(sqrt(56)) = 7 is prime and 7 divides 56.

Programs

  • Mathematica
    Flatten[ #(#+{0, 1, 2})&/@Prime/@Range[20]]
a[n_] := (p=Prime[Floor[n/3+1]])(p+Mod[n, 3])
dpipQ[n_]:=Module[{c=Floor[Sqrt[n]]},PrimeQ[c]&&Divisible[n,c]]; Select[Range[ 4000],dpipQ] (* Harvey P. Dale, Mar 10 2013 *)
  • PARI
    ipsqrt(n) = { sr= 0; for(x=1,n, v = floor(sqrt(x)); if(isprime(v) && x%v == 0, print1(x" "); sr+=1.0/x; ); ); print(); print(sr); } \\ numbers divisible by prime integer parts of their square roots.

Formula

a(n) = prime(floor(n/3+1))*(prime(floor(n/3+1)) + (n mod 3))

A334805 a(n) = Product_{d|n} lcm(d, sigma(d)) where sigma(k) is the sum of divisors of k (A000203).

Original entry on oeis.org

1, 6, 12, 168, 30, 864, 56, 20160, 1404, 16200, 132, 2032128, 182, 56448, 43200, 9999360, 306, 23654592, 380, 190512000, 451584, 313632, 552, 29262643200, 23250, 596232, 1516320, 88510464, 870, 100776960000, 992, 20158709760, 836352, 1685448, 2822400
Offset: 1

Views

Author

Jaroslav Krizek, Jun 26 2020

Keywords

Examples

			a(6) = lcm(1, sigma(1)) * lcm(2, sigma(2)) * lcm(3, sigma(3)) * lcm(6, sigma(6)) = lcm(1, 1) * lcm(2, 3) * lcm(3, 4) * lcm(6, 12) = 1 * 6 * 12 * 12 = 864.

Crossrefs

Cf. A334783 (Sum_{d|n} lcm(d, sigma(d))), A334491 (Product_{d|n} gcd(d, sigma(d))).
Cf. A000203 (sigma(n)), A009242 (lcm(n, sigma(n))), A036690.

Programs

  • Magma
    [&*[LCM(d, &+Divisors(d)): d in Divisors(n)]: n in [1..100]]
  • Mathematica
    a[n_] := Product[LCM[d, DivisorSigma[1, d]], {d, Divisors[n]}]; Array[a, 35] (* Amiram Eldar, Jun 27 2020 *)
  • PARI
    a(n) = my(d=divisors(n)); prod(k=1, #d, lcm(d[k], sigma(d[k]))); \\ Michel Marcus, Jun 27 2020

Formula

a(p) = p^2 + p for p = primes (A000040).
Previous Showing 11-15 of 15 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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